Which of the following is-are true for the equation- a0xn-a1xn-1-a2xn-2-an-0- if S1-S2-S3 - Sn are the sum of the roots taken 1- 2- 3- - n at a time respectively-

For the equation, nbsp; nbsp;a_0x^n+a_1x^n 1+a_2x^n 2+...+a_n=0. If S_1,S_2,S_3, ⋯ S_n etc. are the sum of the roots taken 1, 2, 3, ⋯ n at a time respective ...

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Standard XII

Mathematics

Relation Between Roots and Coefficients for Higher Order Equations

Which of the ...

Question

Which of the following is/are true for the equation, $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n} = 0$ , if $S_{1}, S_{2}, S_{3} \dots S_{n}$ are the sum of the roots taken $1, 2, 3, \dots, n$ at a time respectively?

S_{1} = - \frac{a_{1}}{a_{0}}

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S_{2} = - \frac{a_{2}}{a_{0}}

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S_{n} = (- 1)^{n} \frac{a_{n}}{a_{0}}

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S_{3} = \frac{a_{3}}{a_{0}}

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Solution

The correct option is C $S_{n} = (- 1)^{n} \frac{a_{n}}{a_{0}}$
For the equation, $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n} = 0.$
If $S_{1}, S_{2}, S_{3}, \dots S_{n}$ etc. are the sum of the roots taken $1, 2, 3, \dots n$ at a time respectively then:
$\Rightarrow S_{1} = - \frac{a_{1}}{a_{0}}; S_{2} = \frac{a_{2}}{a_{0}}; S_{3} = - \frac{a_{3}}{a_{0}} \dots S_{n} = (- 1)^{n} \frac{a_{n}}{a_{0}}$

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Which of the following is/are true for the equation, a0xn+a1xn−1+a2xn−2+...+an=0, if S1,S2,S3⋯Sn are the sum of the roots taken 1,2,3,⋯,n at a time respectively?

The correct option is C Sn=(−1)nana0For the equation, a0xn+a1xn−1+a2xn−2+...+an=0. If S1,S2,S3,⋯Sn etc. are the sum of the roots taken 1,2,3,⋯n at a time respectively then: ⇒S1=−a1a0;S2=a2a0;S3=−a3a0⋯Sn=(−1)nana0

Which of the following is/are true for the equation, $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n} = 0$ , if $S_{1}, S_{2}, S_{3} \dots S_{n}$ are the sum of the roots taken $1, 2, 3, \dots, n$ at a time respectively?